Algebraic Branching Programs, Border Complexity, and Tangent Spaces

Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most $k$ is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width. It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most $k$ is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes

Variety Membership Testing in Algebraic Complexity Theory

In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r

Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests

Equivalence testing for a polynomial family {g_m} over a field F is the following problem: Given black-box access to an n-variate polynomial f(x), where n is the number of variables in g_m, check if there exists an A in GL(n,F) such that f(x) = g_m(Ax). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the two popular variants of the iterated matrix multiplication polynomial: IMM_{w,d} (the (1,1) entry of the product of d many w $\times$ w symbolic matrices) and Tr-IMM_{w,d} (the trace of the product of d many w $\times$ w symbolic matrices). The families Det, IMM and Tr-IMM are VBP-complete, and so, in this sense, they have the same complexity. But, do they have the same equivalence testing complexity? We show that the answer is 'yes' for Det and Tr-IMM (modulo the use of randomness). The result is obtained by connecting the two problems via another well-studied problem called the full matrix algebra isomorphism problem (FMAI). In particular, we prove the following: 1. Testing equivalence of polynomials to Tr-IMM_{w,d}, for d$\geq$ 3 and w$\geq$ 2, is randomized polynomial-time Turing reducible to testing equivalence of polynomials to Det_w, the determinant of the w $\times$ w matrix of formal variables. (Here, d need not be a constant.) 2. FMAI is randomized polynomial-time Turing reducible to equivalence testing (in fact, to tensor isomorphism testing) for the family of matrix multiplication tensors {Tr-IMM_{w,3}}. These in conjunction with the randomized poly-time reduction from determinant equivalence testing to FMAI [Garg,Gupta,Kayal,Saha19], imply that FMAI, equivalence testing for Tr-IMM and for Det, and the $3$-tensor isomorphism problem for the family of matrix multiplication tensors are randomized poly-time equivalent under Turing reductions.Comment: 36 pages, 2 figure

A note on VNP-completeness and border complexity

In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, he defined the reduction notion known as p-projection and he proved that the permanent polynomial and the Hamiltonian cycle polynomial are VNP-complete under p-projections. In 2001 Mulmuley and Sohoni (and independently B\"urgisser) introduced the notion of border complexity to the study of the algebraic complexity of polynomials. In this algebraic machine model, instead of insisting on exact computation, approximations are allowed. This gives VNP the structure of a topological space. In this short note we study the set VNPC of VNP-complete polynomials. We show that the complement VNP \ VNPC lies dense in VNP. Quite surprisingly, we also prove that VNPC lies dense in VNP. We prove analogous statements for the complexity classes VF, VBP, and VP. The density of VNP \ VNPC holds for several different reduction notions: p-projections, border p-projections, c-reductions, and border c-reductions. We compare the relationship of the completeness notions under these reductions and separate most of the corresponding sets. Border reduction notions were introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Our paper is the first structured study of border reduction notions.Comment: Theorem 1 has been strengthened. The topology has been adjusted. Section 7 is ne

Descomposiciones en CW-Complejo de Curvas Algebraicas Planas y Fibras de Milnor de Singularidades Cuasiordinarias no Aisladas

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 19/11/2019This is a brief abstract that outlines the topics and contents of this work. The reader interested in a more detailed overview can skip directly to the introduction. The braid monodromy is an invariant of algebraic curves that encodes strong information about their topology. Let C be an affine algebraic plane curve, defined by a polynomial function f, and having a generic projection on the x axis of C². The braid monodromy of C can be presented as a homomorphism ρ : π1(C\{x1, . . . ,xₘ}) → βₙ, where x1, . . . ,xₘ are the values of x on which f(x, y) have multiple roots, and βₙ denotes the braid group of n strands. If we see the curve as the image of a multivalued function g, the image under ρ of a given loop is determined by the paths in C² that (x, g(x)) follows when x runs along the loop. The braid monodromy has a long story and its development and applications has passed through the works of Zariski ([44, 45]), van Kampen ([16]), Moishezon and Teicher ([26, 27, 28, 29, 30, 31]), and Carmona ([9]) among many others ([11, 10, 19, 37, 2, 18, 3]). A result by Carmona ([9]) shows that the braid monodromy of a curve C determines the topology of the pair (P², C). He also provided a program that calculates the braid monodromy of a curve from its equation. However, it remained an open problem to find what this topology actually is. This is, given the braid monodromy of C, to find a description for the topology of (C², C) or (P², C). In this work we provide such a presentation for the affine case. It consists of a regular CW decomposition of the pair (D,C∩D), where D is a large enough polydisc in C². The construction uses the presentation of the braid monodromy in the form of local braids and conjugating braids. In this presentation the local braids must be given as an ordered set of independent sub-braids, associated with different preimages of a critical value of a generic projection. The main theorem concerning the algebraic curves states the good definition of this decomposition (Theorem 1.18)...Este es un breve resumen que describe los temas y contenidos principales de este trabajo. El lector interesado en una descripción más detallada puede saltar directamente a la intriducción. La monodromía de trenzas es un invariante de las curvas algebraicas que codifica fuerte información acerca de su topología. Sea C una curva algebraica afín plana, definida por una función polinómica f, y con una proyección genérica en el eje x de C². La monodromía de trenzas de C puede ser presentada como un homomorfismo ρ : π1(C\{x1, . . . ,xₘ}) → βₙ, donde x1, . . . ,xₘ son los valores de x sobre los cuales f(x, y) tiene raíces múltiples, y βₙ denota el grupo de trenzas de n hebras. Si vemos a la curva como la imagen de una función multivaluada g, la imagen bajo fl de un lazo dado está determinada por los caminos en C² que sigue (x, g(x)) cuando x recorre el lazo. La monodromía de trenzas tiene una larga historia y su desarrollo pasa por los trabajos de Zariski ([44, 45]), van Kampen ([16]), Moishezon y Teicher ([26, 27, 28, 29, 30, 31]) y Carmona ([9]), entre muchos otros ([11, 10, 19, 37, 2, 18, 3]). Un resultado de Carmona ([9]) muestra que la monodromía de trenzas de una curva C determina la topología del par (P2, C). Carmona además proporcionó un programa que calcula la monodromía de trenzas de una curva a partir de su ecuación. Sin embargo, permaneció abierto el problema de determinar en efecto esta topología. Esto es, dada la monodromía de trenzas de C, encontrar una presentación para la topología de (C², C) o (P², C). En este trabajo proporcionamos tal presentación para el caso de curvas afines. La misma consiste en una descomposición CW regular del par (D,C∩D), donde D es un polidisco suficientemente grande en C². La construcción de dicha descomposición utiliza la presentación de la monodromía de trenzas como trenzas locales y trenzas conjugadas. En esta presentación las trenzas locales deben estar dadas como un conjunto ordenado desub-trenzas independientes, asociadas a las diferentes preimágenes de un valor crítico de una proyección genérica. El teorema principal sobre las curvas algebraicas afirma la buena definición de esta descomposición (Teorema 1.18)...Fac. de Ciencias MatemáticasTRUEunpu

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.